High-density scaling identity for the half-space log-Gamma polymer when $\tilde{\beta}<\tilde{t}$

Let $\tilde{\beta}$, $\tilde{t}$, $\alpha$, and $A$ be the parameters from the cited high-density asymptotic theorem, and let $\widetilde{Q}_{\tilde{\beta}<\tilde{t}}^{High}$ be the corresponding limiting function. Let $\widetilde{F}_{-\tilde{\beta},-\tilde{t}}^{HD}$ denote the stationary exponential-LPP limiting function. High-density matching conjecture. The following identity of functions is expected:

$$\left(1+\frac{1}{\tilde{\beta}-\tilde{t}}\partial_s\right)\frac{1}{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}}(A-\mathrm{i}y)e^{-Ax}e^{\mathrm{i}xy}\widetilde{Q}_{\tilde{\beta}<\tilde{t}}^{High}(x+v)\\,dx\\,dy=\widetilde{F}_{-\tilde{\beta},-\tilde{t}}^{HD}(v).$$

The conjecture expresses agreement between the log-Gamma and stationary exponential-LPP asymptotic formulas in the high-density regime. The source does not state that this identity has been proved.